That's a good question. You're right that a random grad student in a lab can't replicate, e.g., the discovery of the Higgs boson!

However, at least for the Higgs (and I believe this is standard for most such large experiments), there were actually two different groups using two different apparatuses, both at CERN.

There was the CMS[0] detector and ATLAS[1] detector, used and analyzed by two different groups of physicists, resulting in two papers[2,3], published on the same day.

[0] https://en.wikipedia.org/wiki/Compact_Muon_Solenoid [1] https://en.wikipedia.org/wiki/ATLAS_experiment [2] https://arxiv.org/abs/1207.7235 [3] https://arxiv.org/abs/1207.7214